Similarly, we may apply the upper-tail bound of Hoeffding's inequality [1]} (Corollary 1.10.13), to obtain:
\(\textnormal {Pr}\left(\sum _{i=1}^{\alpha \beta } {\mathbb {1}_{i}} \le c\sqrt{\beta \log \frac{1}{\delta }} + \mathbf {E}\sum _{i=1}^{\alpha \beta }{\mathbb {1}_{i}}\right) \le \delta /2.\) 
